We  give a  necessary and  sufficient condition  for the  convergence in
distribution of a conditioned  Galton-Watson tree to Kesten's tree. This
yields  elementary proofs  of Kesten's  result  as well  as other  known
results on local limit of conditioned Galton-Watson trees. We then apply
this  condition to  get new  results, in  the critical  and sub-critical
cases, on the limit in  distribution of a Galton-Watson tree conditioned
on having a large number of individuals with out-degree in a given set.